Questions tagged [ap.analysis-of-pdes]
Partial differential equations (PDEs): Existence and uniqueness, regularity, boundary conditions, linear and non-linear operators, stability, soliton theory, integrable PDEs, conservation laws, qualitative dynamics.
4,275
questions
3
votes
1
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Positive first eigenvalue; operator satisfies maximum principle
I am attempting to understand a paper. They have $u$ is a stable smooth solution of $ -\Delta u = f(u)$ in $ \Omega$ with $ u=0$ on $ \partial \Omega$ where $\Omega$ is a bounded domain in Euclidean ...
- 1,363
5
votes
1
answer
307
views
Finding vector fields on $S^2$ with equal divergence
Let $\mathfrak{X}_{CK}^{\perp}$ be the space of vector fields on $S^2$ that are $L^2$-orthogonal to conformal Killing vector fields. Let $\mathfrak{X}_{CK}$ be the 6-dimensional space of conformal ...
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6
votes
1
answer
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Solving $\Delta \text{tr}(h) - \mathrm{div}(\mathrm{div}(h)) + \text{tr}(h) = f$ on $S^2$
$\DeclareMathOperator\ddiv{div}\DeclareMathOperator\tr{tr}\newcommand{\conf}{\mathrm{conf}}$Consider this PDE on a symmetric tensor $h$ on $S^2$:
$$\Delta \text{tr}(h) - \ddiv(\ddiv(h)) + \tr(h) = f$$
...
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0
votes
0
answers
108
views
Hessian estimates of eigenfunctions without Bochner
let $\Omega$ be a bounded domain in a Riemannian manifold $(M,g)$. Consider the Dirichlet eigenvalues and eigenfunctions of Laplacian on $\Omega$, that are, the $\lambda_i>0$ and $\phi_i\in H^{1}_0(...
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2
votes
1
answer
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Does higher volatility of SDE imply lower probability of staying positive?
Given two SDEs $X^1$, $X^2$ :
$$X^i_t=1+t+\int_0^t\sigma_i(s)dW_s,\quad \forall t\ge 0,$$
where $\sigma_i:\mathbb R_+\to [1/2,1]$ are non-decreasing s.t. $\sigma_1(t)\le \sigma_2(t)$ for all $t\ge 0$....
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3
votes
1
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Dirichlet to Neumann operator for a nonlocal ODE
Consider the following nonlocal ODEs on $[1,\infty)$.
#1)
$$\begin{align}
r^2 f''(r) + 2rf'(r)-l(l+1) f(r) &= -\frac{(f'(1) + f(1))}{r^2}\\
f(1) &= \alpha \\
\lim_{r\to \infty} f(r) &= 0
\...
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vote
0
answers
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Existence of $C^{2, \alpha}$ solution to $a^{ij}(x,u,Du)D_{ij}u+b(x,u,Du)=0$ using the Leray–Schauder theorem in "Elliptic PDE" of Q. Han & F. Lin
In this part of the book "Elliptic PDE" of Qing Han & Fanghua Lin, the Leray–Schauder existence theorem is applied to prove the existence of $C^{2, \alpha}(\bar{\Omega})$ solution.
For $\...
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2
votes
1
answer
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Convergence of an infinite sum, whose terms are supported in balls, in Besov space
Suppose we have a ball $B \subset \mathbb R^n$ and $\alpha >0$, and $\{ u_j\}_{j\geqslant-1} $ is a sequence of smooth functions such that the Fourier transforms $\mathcal{F}(u_j) $ are supported ...
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2
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0
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Elliptic pde and boundary layer estimates
I have a question that is related to finding some boundary layer estimates. I am sure there is a general method for this but I don't know it.
I will pose the problem in two dimensions (but here there ...
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2
votes
0
answers
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Existence and uniqueness for $\Delta f + \lambda f = g$ on $S^2$ for $\lambda>0$ [closed]
Consider the PDE
$$\Delta f + \lambda f = g$$
on $S^2$, where $\Delta$ is with respect to the round metric, $g \in L^2(S^2)$ and $\lambda>0$. I wish to study the existence and uniqueness of this ...
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0
answers
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Solve a coupled PDE in a rectangle
We consider a coupled PDE in a rectangle $\Omega=(-1,1)\times(-1,1)$. For the simplicity, we assume that the functions are periodic in $x_{1}$ direction.
\begin{cases}
\nabla\cdot u=f_{1},\ & \...
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votes
0
answers
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Convergence of a infinite sum in Besov space
Suppose we have an annulus $A \subset \mathbb R^n$, which is the set $\{x|0<r \leqslant\|x\| \leqslant R\}$, $\alpha \in \mathbb R$ and $\{ u_j\}_{j\geqslant-1} $ be a seqence of smooth functions ...
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votes
3
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Is $p_t(y,x)$ the kernel of the time reversal of the diffusion $X$, for $p_t(x,y)$ the kernel of $X$?
Short version. If $X$ is a diffusion with generator $L$ and the Lebesgue measure is invariant for $X$, then $L^*$ has no term of order zero and it corresponds to another diffusion $X^*$. Denoting by $...
- 3,049
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Convergence of Sobolev traces over a sequence of boundaries
Let $u$ be a function in the Sobolev space $W^{\frac 3 2, 2}(\mathbb R^n)$.
Let $\Sigma\subset \mathbb R^n$ be the graph of a Lipschitz function $f:B_{n-1}(0,1) \subset \mathbb R^{n-1} \rightarrow \...
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2
votes
0
answers
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"Equivalent" reference to "Quelques méthodes" by J-L. Lions
I've just started learning about some methods to deal with parabolic equations, and in a lot of papers they refer to the book "Quelques méthodes de résolution des problèmes aux limites non ...
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1
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220
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A proof for the existence of smooth solution of PDE in form $\Delta u=f(x, u)$ in Michael E. Taylor's book Partial Differential Equations III
This part is from page 107 in Michael E. Taylor's book Partial Differential Equations III.
In this part, we want a proof for the existence of smooth solution of the PDE
$\Delta u=f(x, u)$ on $U$ with ...
- 729
3
votes
1
answer
271
views
Definition of Martin kernels
Let $\Omega \subset \mathbb{R}^n$ $(n \ge 3)$ be a bounded $C^{1,1}$ domain and let $X$ be a Markov process in $\Omega$. My question is regarding the existence of the Green function and Martin kernel ...
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votes
2
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318
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Compressible Ebin-Marsden?
In Ebin and Marsden's paper Groups of Diffeomorphisms and the Motion of an Incompressible Fluid, there is a footnote on the first page indicating that non-homogeneous cases and the case of ...
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0
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119
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A problem about using the moving plane method to prove radial symmetry of the $C^{2}$ global solution of a elliptic PDE in $R^{2}$
Recently I'm learning the use of moving plane method to prove radial symmetry of $C^{2}$ global solution of a PDE in $R^{2}$, and I'm reading a paper where this method is applied: precisely I'm ...
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1
vote
1
answer
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How to prove that the L-infinity norm is smaller than the Besov norm?
Suppose we have a distribution $u\in B_{\infty,\infty}^\alpha$, the Besov space with regularity coefficient $\alpha>0$. How to prove the folowing inequality?
$$
\|u\|_{L^\infty}\leqslant c\|u\|_{B_{...
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6
votes
1
answer
311
views
On the fundamental solution for elliptic PDE
In the well known paper by Littman-Weinberger-Stampacchia "Regular points for elliptic equations with discontinuous coefficients", the authors were able to prove the validity the following ...
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1
vote
1
answer
111
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Prove $\int_\Omega \left(\rho_{1} \ln \frac{\rho_{1}}{\rho_{2}}\right)dx dy \leq C\int_\Omega |\rho_1-\rho_2|dxdy$ for $0 \le \rho_1, \rho_2 \in L^1$
Let $\rho_1, \rho_2 \in L^1(\Omega;\mathbb R_+)$ such that $\int \rho_i|\ln \rho_i| < \infty$. Is it true that there exists a constant $C>0$ such that
\begin{align*}
\int_\Omega \left(\rho_{1} \...
user140746
2
votes
0
answers
113
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Some problems about energy estimates of elliptic equation
Recently I am reading a book of elliptic equations. In the beginning there is a famous Caccioppli inequality for weak solutions. The theorem is stated as follows
Suppose that $ u\in H^1(B(0,1)) $ ...
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4
votes
1
answer
299
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Taylor coefficients of Hadamard product
I imagine this to be a very classical question in complex analysis:
Consider the Hadamard product
$$g(\mu) = \prod_{n=1}^{\infty}E_1(\mu z_n),$$
where $E_1(z):=(1-z)e^z$ is the first elementary ...
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2
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0
answers
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A regularity result for semilinear PDE of the form $\Delta u=f(x, u)$ in Michael E. Taylor's book "Partial Differential Equations III"
Let $M$ be a bounded domain in $\Bbb R^2$: under the assumption that
$$
\partial_{u} f(x, u)=0 \text { for }|u| \geq K\label{1}\tag{1.6}
$$
Michael E. Taylor said that (proposition (1.3))
For $k=1,2, \...
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4
votes
1
answer
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Spectrum Cauchy-Euler operator
A Cauchy-Euler operator is an operator that leaves homogeneous polynomial of a certain degree invariant, named after the Cauchy-Euler differential equations
We consider the operator
$$(Lf)(x) = \...
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2
votes
1
answer
76
views
Parabolic system with coupling in the diffusion
Let's consider the parabolic system
$$
\begin{cases}
u_t - \Delta u -a\Delta(uv) = 0 \\
v_t - \Delta v - b\Delta(uv) = 0
\end{cases}
$$
with $a,b >0$. What is the name of this system? Are there ...
user139844
1
vote
0
answers
172
views
Converting equation on the sphere to $\mathbb{R}^n$
Consider the following equation $-\Delta_{\mathbb{S}^n} u = u$ where the $(\mathbb{S}^n,g)$ where $g$ is the usual metric induced by the inverse stereographic projection $S:\mathbb{R}^n\to \mathbb{S}^...
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7
votes
1
answer
318
views
Higher regularity of solutions of non-linear elliptic PDE
Let $\Omega\subset \mathbb{R}^n$ be a bounded domain with infinitely smooth boundary. Let $u\in C^2(\bar \Omega)$ be a solution of the Dirichlet problem for the non-linear equation
\begin{eqnarray}
F(...
- 21.1k
5
votes
1
answer
158
views
Asymptotics for repulsive aggregation(-diffusion) equation
Consider the aggregation-diffusion equation
$$
\frac{\partial \rho}{\partial t} = \nabla (\rho \nabla(W\star \rho)) + \nu \Delta \rho,
$$
where $W:\mathbb{R}^d \to \mathbb{R}$ is a twice continuously ...
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2
votes
0
answers
50
views
Rescaling an elliptic system
I have a basic question regarding rescaling an elliptic system when trying to get apriori estimates.
Consider an elliptic system say of the form
$$-\Delta u(x) = u^{p_1} v^{q_1}, \quad -\Delta v = u^{...
- 1,363
8
votes
1
answer
594
views
Calderon-Zygmund decomposition on manifolds?
The classical Calderon-Zygmund decomposition says that if $f\geq 0$ is $L^1$ on a cubes $B$, with average value $\alpha$, then there is a sequence of disjoint cubes $B_j$, such that the average of $f$ ...
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3
votes
0
answers
73
views
An exact solution to a homogeneous linear second order differential eq. with variable coefficients
Our differential equation
Or in words: $$ \Bigg[\frac{\partial^2}{\partial u^2} +\frac{h_+^2 \omega \sin(\omega u) \cos(\omega u)}{1-h_+^2\cos^2(\omega u)} \frac{\partial}{\partial u} + \frac{k_x^...
5
votes
0
answers
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Are there connected closed 4-manifolds admitting a regular Almost Lagrangian distribution, and which are not Lorentzian?
In the category of real differential manifolds, connected (of $ C ^ {\infty} $ class in the sequel), closed of dimension 4, is there any manifold admitting a regular Almost Lagrangian distribution and ...
2
votes
1
answer
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Simple elliptic pde problem
I have a question that is clearly not research level, but it's confusing me so I will ask anyway.
There must be some little logic flaw I am missing. Take $\Omega$ a bounded smooth domain in $\mathbb ...
- 1,363
5
votes
1
answer
236
views
Parabolic equation with Cauchy boundary condition
Consider the domain $[0,1] \times [0,T]$ and the uniformly parabolic operator $L -\partial_t$ with smooth coefficient. I would like to obtain the existence of the problem
\begin{equation}
\left\{\...
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votes
0
answers
332
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Sobolev inequality on the sphere derivation
I am reading the following paper (preprint here) and the author starts by stating the Sobolev inequality on the Sphere $\mathbb{S}^d$
$$\frac{p-2}{d}\int |\nabla u|^2 + \int |u|^2 \geq \left(\int |u|^...
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2
votes
1
answer
137
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Elliptic equation on square with periodic boundary values for the solution and it's partial derivatives
Suppose that $\Omega=[0,1]^2$. I will say that a real valued function $u$ on $\Omega$ satisfies periodic boundary values if
$$u(x,0)=u(x,1), \;u(0,y)=u(1,y),\;\;\;\text{ for all }x,y\in[0,1].$$
Now ...
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votes
0
answers
66
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Regularity of solution to Cauchy problem given regular initial data
Let $f\in L^2_1([0,T]\times \mathbb{T}^m)$ (Sobolev space of maps of regularity $1$, $\mathbb{T}^m$ is the $m$-dimensional torus)
be a solution of a Cauchy problem
$$\frac{d}{dt} f(t) = A f(t)$$
$$f(0)...
- 2,523
1
vote
0
answers
95
views
Explicit expression for the Poisson kernel solving the Dirichlet problem for geodesic balls
Let $X$ be a Riemannian manifold with the Laplace-Beltrami operator denoted by $\mathscr L$ and we look at its geodesic balls say $B$. Let $u$ be a continuous function on the geodesic sphere which is ...
- 201
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votes
1
answer
433
views
General validity of separation of variables
Let $L$ be any differential operator (not necessarily linear).
Given initial conditions and boundary conditions (of any type), I am interested in general statements of the form:
Given a boundary ...
- 237
2
votes
1
answer
92
views
How to estimate higher order regularity for wave type equation with time dependant coefficients?
Consider the following wave-type equation,
$$u_{tt}-\frac{2}{t}u_t-\Delta u=g(t,x)$$
where $(t,x)\in [\epsilon, 1]\times \mathbb{R}^3$ for some $\epsilon>0.$ Furthermore assume that $(u,u_{t})=(0,0)...
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3
votes
1
answer
137
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Gluing of two solutions to the same parabolic equation
Consider the domain $[0,1] \times [0,T]$ and the uniformly parabolic operator $L -\partial_t$ with smooth coefficient. Suppose I have $u_1(x,t) \in C^\infty([0,1] \times [0,T])$ solving
\begin{...
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4
votes
0
answers
144
views
What role do semiclassical methods play in the study of Ginzburg--Landau-type equations?
As far as I understand, semiclassical limits are used in quantum mechanics to analyse equations that depend on a small parameter $\hbar$. Apparently studying properties of the PDE as $\hbar \to 0$ ...
- 4,968
2
votes
1
answer
241
views
Compactness for initial-to-final map for heat equation
Let $M$ be a compact smooth manifold without boundary. Let $T>0$ and let $g$ be a smooth Riemannian metric on $M$. Given any $f \in L^2(M)$ let $u$ be the unique solution to the equation
$$\...
- 4,111
1
vote
1
answer
378
views
Rewriting PDE as "push-forward"
Suppose that we have the following PDE
$$\partial_t \mu_t = \nabla\cdot \left(\nabla \mu_t - (b*\mu_t)\mu_t\right), \tag{1}$$
with $\mu_0$ being a (smooth) probability measure/density on $\mathbb{R}^d$...
- 700
1
vote
0
answers
117
views
Reference for global theory of Schrödinger operators
Question. What is a good reference to learn about the spectral properties of Schrödinger operators in $\mathbf{R}^n$? I am specifically interested in references that discuss examples where the ...
- 4,968
0
votes
0
answers
66
views
Measure and other properties of nodal domains of Laplacian
Let $(\phi_k,\lambda_k)$ be the couple of eigenfunctions and eigenvalues of the the Laplacian operator on $\Omega \subset \mathbb R^n$.
The nodal set of $\phi_k$ is the set $$\mathcal N_k = \{x \in \...
- 161
0
votes
1
answer
120
views
Iterated integrations by parts using the fractional Laplacian
Let $u \in C^\infty_c(\mathbb{\Omega})$ and $\varphi$ be an eigenfunction of the fractional Laplacian $(-\Delta)^s$ in $\Omega$ with eigenvalue $\lambda$. In what sense, if any, is it true that
$$\...
- 819
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votes
0
answers
90
views
Biharmonic operator and maximum principle (PPP)
I have a question related to the Positivity Preserving Principle (PPP) for $ \Delta^2$ and related topics. Recall if $u$ solves
$$\Delta^2 u = f(x) \mbox{ in } \Omega, \quad u=\partial_\nu u =0 \...
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